Flip Distance and Triangulations of a Ball

TL;DR

This paper investigates the relationship between flip distance of triangulations and tetrahedral counts in polyhedra, showing they can differ significantly with ratios approaching 1.5.

Contribution

It provides counterexamples demonstrating that flip distance and tetrahedral count are not always equal and can differ arbitrarily close to 3/2.

Findings

Flip distance and tetrahedral count can differ significantly.

The ratio between these two measures can approach 3/2.

Counterexamples show they are not always equal.

Abstract

It is known that the flip distance between two triangulations of a convex polygon is related to the minimum number of tetrahedra in the triangulation of some polyhedron. It is interesting to know whether these two numbers are the same. In this work, we find examples to show that the two numbers are different in nature, and their ratio can be arbitrarily close to $\frac{3}{2}$.